MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  imass1 Structured version   Visualization version   GIF version

Theorem imass1 6121
Description: Subset theorem for image. (Contributed by NM, 16-Mar-2004.)
Assertion
Ref Expression
imass1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))

Proof of Theorem imass1
StepHypRef Expression
1 ssres 6023 . . 3 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
2 rnss 5952 . . 3 ((𝐴𝐶) ⊆ (𝐵𝐶) → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
31, 2syl 17 . 2 (𝐴𝐵 → ran (𝐴𝐶) ⊆ ran (𝐵𝐶))
4 df-ima 5701 . 2 (𝐴𝐶) = ran (𝐴𝐶)
5 df-ima 5701 . 2 (𝐵𝐶) = ran (𝐵𝐶)
63, 4, 53sstr4g 4040 1 (𝐴𝐵 → (𝐴𝐶) ⊆ (𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wss 3962  ran crn 5689  cres 5690  cima 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701
This theorem is referenced by:  predrelss  6359  vdwnnlem1  17028  dprdres  20062  imasnopn  23713  imasncld  23714  imasncls  23715  utoptop  24258  restutop  24261  ustuqtop3  24267  utopreg  24276  metustbl  24594  imadifxp  32620  gsumfs2d  33040  esum2d  34073  eulerpartlemmf  34356  bj-imdirco  37172  brtrclfv2  43716  frege97d  43741  frege109d  43746  frege131d  43753  hess  43769  resimass  45183  setrecsss  48931
  Copyright terms: Public domain W3C validator